12.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08b1be3e-2d9a-4832-b230-d5519540f494-40_814_713_219_612}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Figure 4 shows a sketch of part of the curve \(C\) with equation
$$y = x ^ { 3 } - 9 x ^ { 2 } + 26 x - 18$$
The point \(P ( 4,6 )\) lies on \(C\).
- Use calculus to show that the normal to \(C\) at the point \(P\) has equation
$$2 y + x = 16$$
The region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the normal to \(C\) at \(P\).
- Show that \(C\) cuts the \(x\)-axis at \(( 1,0 )\)
- Showing all your working, use calculus to find the exact area of \(R\).